To appear, Artificial Intelligence, 1995
Abduction to Plausible Causes:
An Event-based Model of Belief Update
Craig Boutilier
Department of Computer Science
University of British Columbia
Vancouver, British Columbia
CANADA, V6T 1Z4
email: [email protected]
January 11, 1995
Abstract
The Katsuno and Mendelzon (KM) theory of belief update has been proposed as a reasonable
model for revising beliefs about a changing world. However, the semantics of update relies on
information which is not readily available. We describe an alternative semantical view of update
in which observations are incorporated into a belief set by: a) explaining the observation in terms
of a set of plausible events that might have caused that observation; and b) predicting further
consequences of those explanations. We also allow the possibility of conditional explanations.
We show that this picture naturally induces an update operator conforming to the KM postulates
under certain assumptions. However, we argue that these assumptions are not always reasonable,
and they restrict our ability to integrate update with other forms of revision when reasoning about
action.
Some parts of this report appeared in preliminary form as “An Event-Based Abductive Model of Update,” Proc. of
Tenth Canadian Conf. on in AI, Banff, Alta., (1994).
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To appear, Artificial Intelligence, 1995
1
Introduction
Reasoning about action and change has been a central focus of research in AI for many years, dating
at least to the origins of the situation calculus [20]. For example, a planning agent must be able to
predict the effects of its actions on the world in order to verify whether a potential plan achieves a
desired goal. Actions effect changes in the world, and agents must be able to modify their beliefs
about the world to reflect such considerations. Furthermore, an agent situated in a dynamic world
must be able to reason about changes in the world not simply due to its own actions, but due to the
occurrence of exogenous events as well.
One of the most influential theories of belief change has been the AGM theory proposed by
Alchourrón, Gärdenfors and Makinson [1]. Imagine an agent possesses a belief set or knowledge base
KB. The AGM theory provides a set of postulates constraining the possible ways in which the agent
can change KB in order to accommodate a new belief A. Notice that this revision of KB need not be
straightforward, for the new belief A may conflict with beliefs in KB. It was pointed out by Winslett
[27] that the AGM theory is inappropriate for reasoning about changes in belief due to the evolution
of a changing world. A new form of belief change dubbed update was proposed in full generality by
Katsuno and Mendelzon [16], who provided a set of postulates, distinct from the AGM postulates,
that characterize this type of belief change.
Semantically, Katsuno and Mendelzon have shown that belief update can be characterized by
positing a family of orderings over possible worlds, with each ordering being indexed by some world.
The ordering associated with a specific world can be viewed intuitively as describing the most plausible
ways in which that world can change. To update a knowledge base KB with some proposition A, the
worlds admitted by KB are each updated by finding the most plausible change associated with that
world satisfying A (we describe this formally below). As a concrete example, suppose that someone
observes that the grass in front of her house is wet. She is not sure whether she left her book outside
on the patio, but concludes that if the book is outside it is wet too. There are two possible worlds
admitted by her knowledge, O and
O (the book is outside or it is not).
When the first possibility is
updated with the observation of wet grass, a wet book is the result. When the second possibility is
updated, the book remains dry. The conditional belief O
W (the book is wet only if it was outside)
is part of our agent’s updated belief set.
In this paper, we present an abductive model of belief change suitable for updating beliefs in
response to a changing world. While our semantics induces a class of belief change operators that is
somewhat more general than Katsuno-Mendelzon (KM) update operators, the most compelling aspect
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To appear, Artificial Intelligence, 1995
of our model is the fact that it breaks the KM semantics into smaller, more primitive parts. We argue
that such a model provides a more natural perspective on belief update in response to changes in the
world, and exploits information that is more readily available or easily obtainable from users of a
system. In the following, we use the term update to describe any process of belief change used to
capture changes in belief due to change in the world (not simply those models conforming to the KM
postulates).
In general, we take update to be a two stage process of explanation followed by prediction:
first, an agent explains an observation by postulating some plausible event or events that could have
caused that observation to hold, relative to its initial state of knowledge; second, an agent predicts the
(further) consequences of these events, relative to this initial state. In our example, there are several
possible causes of wet grass, among them the sprinkler turning on automatically, or rain. If rain is
the most plausible of these causing events, our agent concludes that everything on the patio is wet,
including the book if it is out there. Had sprinkler been the most plausible explanation, a different
conclusion would have been reached: the book would be dry regardless of its location. It is these
considerations that allow an agent to determine just what changes in the world are most plausible.
Intuitively, information about the effects of events, as well as their relative plausibility, will be more
readily available or easier to assess than a direct ordering of plausibility over possible “evolutions”
of the world.
We formalize this notion in an abstract manner obtaining a class of explanation-change operators
that are similar in spirit and intent to KM update operators, but somewhat more general. We note that
explanation has often been closely linked with belief revision [12]. Indeed, Boutilier and Becher [5]
present a model of abduction where explanations are determined by explicit belief revision. Given
this connection and the fact that update can be viewed as an essentially abductive process, we may
also take update to be a certain kind of belief revision. This stands in stark contrast with the accepted
wisdom that update and revision are orthogonal forms of belief change. While we could cast our
model as a form of belief revision, this would detract from the main point of the paper. However, we
do elaborate on this connection in the concluding section.
In Section 2 we review the KM postulates for belief update and the KM semantics. In Section 3
we analyze this semantics more closely, and break it into more basic elements. We describe our
abductive view of update and show its relationship to the KM model. In particular, we show that
certain semantic assumptions naturally give rise to the KM theory; however, we argue that these
assumptions are inappropriate as general update principles. We also briefly describe and characterize
a special class of update operators. In Section 4, we analyze our model more deeply and discuss the
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To appear, Artificial Intelligence, 1995
connections to belief revision. We also argue that proper modification of belief states in response to
observations in dynamic settings involves a combination of belief revision and belief update. Finally,
we compare our construction to the model of update proposed by del Val and Shoham [8]. Proofs of
the main results can be found in the appendix.
2
The Semantics of Update
Katsuno and Mendelzon [16] have proposed a general characterization of belief update. Update
is distinguished from belief revision conceptually by viewing update as reflecting belief change in
response to changes in the world, whereas revision is thought to be more appropriate for changing
(possibly erroneous) beliefs about a static world. Update is described by Katsuno and Mendelzon
with a set of postulates constraining acceptable update operators and a possible worlds semantics,
both of which we review here.
We assume the existence of some knowledge base KB, the set of beliefs held by an agent about
the current state of the world. We take our underlying logic to be propositional, based on a finitely
generated language LCPL . We use W to denote the set of possible worlds (or models) suitable for this
language.
If some new fact A is observed in response to some (unspecified) change in the world (i.e., some
action or event occurrence), then the formula KB A denotes the new belief set incorporating this
change. The KM postulates [16] governing admissible update operators are
(U1) KB A j= A
(U2) If KB j= A then KB A is equivalent to KB
(U3) If KB and A are satisfiable, then KB A is satisfiable
(U4) If j= A B then KB A KB B
(U5) (KB A) ^ B
= KB (A B )
j
^
(U6) If KB A j= B and KB B
= A then KB A
j
KB B
(U7) If KB is complete then (KB A) ^ (KB B ) j= KB (A _ B )
(U8) (KB1 _ KB2 ) A (KB1 A) _ (KB2 A)
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To appear, Artificial Intelligence, 1995
A better understanding of the mechanism underlying update can be achieved by considering the
possible worlds semantics described by Katsuno and Mendelzon, which they show to be equivalent
to the postulates. For any proposition
A, let A
k
k
denote the set of worlds satisfying
A.
Clearly,
KBk represents the set of possibilities we are prepared to accept as the actual state of affairs. Since
k
O is the result of some change in the actual world, we ought to consider, for each
possibility w
KB , the most plausible way (or ways) in which w might have changed in order to
make O true. We will call such a change in any world an “evolution” of that world. To capture this
observation
2 k
k
intuition, Katsuno and Mendelzon postulate a family of preorders
fw
:w
2
W
g
where each w is a reflexive, transitive relation over W . We interpret each such relation as follows:
if u w
v then u is at least as plausible a change relative to w (or an evolution of w) as is v . Finally, a
faithfulness condition is imposed: for every world w, the preorder
has w as a minimum element;
that is, w < v for all v = w. Intuitively, this ensures that w is itself more plausible than any other
evolution of w.1
Naturally, the most plausible candidate changes in w that result in O are those worlds v satisfying
O that are minimal in the relation . The set of such minimal O-worlds for each relation , and
each w
KB , intuitively capture the situations we ought to accept as possible when updating KB
with O. In other words,
[ min v : v = O
KB O =
w
6
w
w
2 k
w
k
k
f
k
w
2kKBk
where minw X is the set of minimal elements (w.r.t.
f
j
gg
w
w
) within X . Katsuno and Mendelzon show
that such a formulation of update captures exactly the same class of change operators as the postulates;
thus, we can treat this as an appropriate semantics for the KM update theory.
As an example, consider the following scenario illustrating the application of the KM update
semantics to database update. We know certain facts about an employee Fred: his salary is $40,000,
his job classification is level N , and so on. But, we are unsure whether he works for the Purchasing
department or the Finance department. Thus, our KB admits two possibilities, w and v , reflecting
this uncertainty (see Figure 1). If the orderings
w
and
v
are as indicated in the figure, then
KB updated with the fact that Fred’s salary is $50,000 contains, among other things, the facts
Dept(P) _ Dept(F), Dept(P)
1
Level(N) and Dept(F)
Katsuno and Mendelzon use the term persistent to describe such orderings.
5
Level(N+1). This is due to
To appear, Artificial Intelligence, 1995
Dept(P)
Sal(40)
Lev(N)
Dept(P)
Sal(50)
Lev(N)
Dept(P)
Sal(50)
Lev(N+1)
w
w’
w’’
Dept(F)
Sal(40)
Lev(N)
Dept(F)
Sal(45)
Lev(N)
Dept(F)
Sal(50)
Lev(N+1)
v
v’
v’’
KB
Figure 1: An Update Model
the fact that the closest world to w with the new salary is w0 , while the closest to v is v 00; hence, KB is
determined by the set of worlds fw0; v 00g. This may reflect the fact that such a raise comes only with
a promotion in Finance, whereas promotions are rare and raises more frequent in Purchasing.
The KM semantics shows very clearly one of the main distinctions between update and belief
revision. In belief revision (e.g., using the AGM theory), if an observation O is consistent with KB,
then the revised KB O must be equivalent to KB [ fOg. This needn’t be the case for update. Given
KB as above, we may receive an update transaction
O
(Dept(P)
Sal(40) ) ^ (Dept(F)
Sal(50) )
While KB [ fOg entails Dept(P), and is captured semantically by the set fwg, KB O corresponds
to the set
w; v 00
f
g
and does not commit Fred to a particular department. The crucial distinction is
update’s willingness to consider the evolution of each possible world individually. Belief revision
only considers the belief set KB as a whole.
3
3.1
Update as Explanation
Plausible Causes of Observations
The orderings upon which update semantics are based are interpreted as describing the most plausible
manner in which a world might change. Given the role of update, this interpretation seems correct:
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To appear, Artificial Intelligence, 1995
worlds closer to w in the ordering w are somehow more plausible states into which w might evolve.
It seems reasonable then to update a KB by considering those most plausible changes. In our example
above, if Fred is in Purchasing (world w), then a change of salary of this type is more likely to come
without a change in rank (w0 ) than with a change in rank (w00).
While reasonable, it begs the question: why would one change be judged more plausible than
another? Intuitively, it seems that there are certain events or actions that would cause a change in
w, and that those leading to w0 are more plausible than those leading to w00. For example, the event
RAISE might be more probable than the event PROMOTION (at least, in Purchasing).
Given an observation Sal(50000) — in this case an update transaction — an agent might
come to believe Dept(P)
Level(N) (as we have in our example) as follows: assuming
Dept(P), the most plausible event that might cause such a change in salary is RAISE (rather than
PROMOTION). Thus RAISE is the best explanation for the observation. Adopting this explanation
has, as a further consequence, that job rank (and department) stays the same; thus, belief in Level(N)
remains. In contrast, RAISE (to $50,000) is less likely than PROMOTION in the Finance department.2
Thus, PROMOTION is the most plausible explanation for the observation, which has the additional
consequence Level(N+1). Thus, the two beliefs Dept(P)
Level(N) and Dept(F)
Level(N+1) hold in the updated belief state.
This leads to a very different view of update. When confronted with an observation or update O,
an agent seeks an explanation of O, in terms of some external event that would have caused O had it
occurred.3 While many events might explain O in this way, some will be more plausible than others,
and it will be those the agent adopts. Given such an explanation, one may then proceed to predict
further consequences of these events, and produce the set of beliefs arising from the observation.
With this point of view, the essence of update is captured by a two-step process: a) explanation of
the observation in terms of some event(s); and b) prediction of the (additional) consequences of that
event. We do not presume that the agent has direct knowledge of the event occurrence. If such
direct knowledge is available the problem becomes much simpler, for the agent can simply predict the
effects of this event using some theory of action. This is a very specific update problem, restricting
an agent to updating by observations of the form “Event E occurred.” No explanation is required.4
2
In our example, we assume that a raise to $45,000 is most likely (world v0 ), but that a higher raise is unlikely without a
promotion.
3
In this paper we will usually think of (external) events as the impetus for change, rather than actions over which the
agent has direct control (or of which the agent has direct knowledge).
4
This assumption is embodied to a certain extent in the update models of del Val and Shoham [8, 9] and Goldszmidt and
Pearl [13], as we discuss in Section 4.
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To appear, Artificial Intelligence, 1995
Before formalizing this idea, it is important to realize that this perspective is very natural. It is
reasonable to suppose that an agent (or builder of a KB) has ready access to some description of the
preconditions and effects of the possible events in a given domain. This assumption underlies all work
in classical planning and reasoning about action, ranging from STRIPS [10] to the situation calculus
[20, 24] to more sophisticated probabilistic representations [18, 7]. With such information, the
predictions associated with explanations (event occurrences) can be easily determined. Furthermore,
an ordering over the relative likelihood of possible events also seems something which an agent or
system designer or user might easily postulate. This should certainly be easier to construct than a
direct ordering over worlds according to their likelihood of “occurring.” Indeed, we will show that
such an ordering over worlds is derivable from this more readily available information.
This provides a possible interpretation of the update process, and in our view, a very natural one.5
Furthermore, as we describe in the concluding section (and in detail in [2]), by breaking update into
two components, we will be able to extend the type of reasoning about action one can perform in this
setting.
Using explanation for reasoning about action has been proposed by a number of people, especially
within the framework of the situation calculus. Work on temporal projection and prediction failures
often exploits the notion of explanation. For instance, Morgenstern and Stein [21] propose a model
where an observation that conflicts with the predicted effects of an agent’s actions causes the agent to
infer the existence of some external event occurrence. Shanahan [26] proposes a model with a similar
motivation, but adopts a truly abductive model (where candidate events are hypothesized rather than
deduced from an observation). Our model will be rather different in several ways. First, explanations
will be conditional (i.e., explaining events are conditioned on certain propositions). Second, the
criteria used for adopting explaining events will be based on the relative plausibility of events. Third,
we will not limit attention to any particular model of action (such as the situation calculus). Finally,
our goal is to show how explanation can account for the update of a knowledge base. We should point
out that Reiter [25, and personal communication] has informally suggested that update can be viewed
as explanation to events causing an observation. We will proceed to show that this is, in fact, the case.
5
This should not be taken as a criticism of update for requiring that a reasoning agent have an explicitly specified family
of preorders at its disposal. One can reason about update with syntactic constraints or by any other means. The point is that,
from a semantic point of view, the preorders and syntactic constraints seem to be induced by considerations about action
effects and plausible event occurrences.
8
To appear, Artificial Intelligence, 1995
3.2
A Formalization
To capture update in terms of explanation, we require two ingredients missing from the KatsunoMendelzon account: a set of events that cause changes, and an event ordering that reflects the relative
plausibility of different event occurrences.
We assume a finitely generated propositional language with an associated set of worlds W . Let
E be a finite event set, the elements of which are primitive events. In general, e E is a mapping
e : W 2 . For w W and e E , we use e(w) to denote the result of event e occurring in world
w. This is a set of worlds, each of which is a possible outcome of e occurring at w. An event with
more than one possible outcome is nondeterministic. A deterministic event is any e E such that
e(w) is a singleton set for each w W . A deterministic event set is an event set all of whose events
are deterministic. We assume that events are total functions on the domain W , so that every event
can be applied to each world. In addition, we insist that e(w) = for each e; w.6 We emphasize that
not only are all possible outcomes of an event captured by the set e(w), but also that each world in
e(w) is a legitimate, plausible outcome.
2
!
W
2
2
2
2
6
;
Typically, events are not specified as mappings of this type. Rather, for each event (or action), a
list of conditions are provided that influence the outcome of the event. For each such condition, a set
of effects is specified. An example of this is the classical situation calculus representation of actions
(in the deterministic case). Another is the modified STRIPS representation presented in [18, 6]. The
key feature of these, and other representations, is that each action/event induces a function between
worlds (or worlds and sets of worlds).7 Thus, most action representations will fit within this abstract
model. While we do not delve into the representation of actions, our examples will suggest ways in
which traditional representations can be augmented with the features of our model.
As a further generalization, if events are nondeterministic, we might suppose that the possible
outcomes are ranked by probability or plausibility. We set aside this complication (but see [2]).
In order to explain certain observations by appeal to plausible event occurrences, we need some
metric for ranking explanations. We assume that the events in the set E are ranked by plausibility;
6
It is best to think of events as analogous to “action attempts.” If the preconditions for the “successful” occurrence
of the event are not true at a given world, then the effects can be null, or unpredictable or something like that. Allowing
preconditions is a trivial and uninteresting addition for our purposes here.
7
In the case of the situation calculus, dynamic logic or other temporal formalisms, one would require some solution to
the frame problem. For example, the solution of Reiter [24] induces just such a mapping.
9
To appear, Artificial Intelligence, 1995
hence, we postulate an indexed family of event orderings
fw
over E . We take e w
world w.8
We require that
:w
2
W
g
f to mean that event e is at least as plausible (or likely to occur) as event f in
w
be a preorder for each
w, and will occasionally assume that
w
is a total
preorder. Once again, we do not expect that this family of orderings will be presented explicitly.
Compact representation schemes are possible. For example, in our database example we might
suppose that a user can specify the constraint that a RAISE event is more plausible than a PROMOTION
event for employees of the Purchasing department. The relative plausibility need not be asserted
explicitly for each world satisfying Dept(P).
We note that there are few restrictions on the relative plausibility of events in any given ordering
w
. The only structural basis for the logical comparison of events is through outcome sets, but
these provide no logical constraints on relative plausibility. If we have two events e and f such that
e(w) f (w), we impose no constraints on the relative ordering of e and f in . In particular, we
cannot insist that an event e with fewer possible outcomes be judged more likely than an event f .
w
For instance, imagine two events, flipping a coin and placing a coin, such that flipping results in two
possible outcomes (heads, tails) and placing has three outcomes (heads, tails, edge). This provides no
a priori reason to consider flipping or placing more likely than the other.
Putting these ingredients together, we have the following definitions:
Definition An event model is a triple
(mappings e : W
!
W; E;
h
, where
i
W
is a set of worlds,
2W ) and is an indexed family of events orderings fw
each w is a preorder over E ).
Definition A deterministic event model is an event model where every e
for all
w W , e(w) = v
2
E is a set of events
: w W (where
f g
for some v
2
W ).
2
2
g
E is deterministic (i.e.,
A total order event model is an event model
where each event ordering w is a total preorder over E .
Given an event model, an agent is able to incorporate a new piece of information through a process
of explanation and prediction as discussed above. An explanation of an observation is some event e
8
Other models of event orderings are possible, including using a fixed ordering for all worlds, or associating event
plausibility with belief sets (or sets of worlds) rather than individual worlds. However, these seem less compelling.
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To appear, Artificial Intelligence, 1995
that, when applied to the world under investigation, possibly causes
O.
However, the agent should
be interested only in the most plausible such events.
Definition Let O be some proposition and w
is
2
W . The set of weak explanations of O relative to w
e E : e(w)
Expl(O; w) = min
f
2
O =
\ k
w
An event e is a weak explanation of O relative to w iff e
2
we say that O is unexplainable relative to w.
k 6
;g
Expl(O; w). If Expl(O; w) =
,
;
In other words, e explains O in a world w just when there is some possible outcome of e that satisfies
O, and no more plausible event e0 has this feature. Such explanations are called weak explanations
because, before the observation O is made, an agent would not, in general, be able to predict that O
would result from e. The agent merely knows that O is true of some possible outcome. This is often
the most we can expect in a domain with nondeterministic events. For example, someone tossing a
coin onto a chess board is a quite reasonable explanation for the fact that the coin is on a black square;
but knowing the event occurred is not enough to predict that outcome, for it might well have landed
on a white square.
A predictive explanation is similar, but we insist that each outcome of e satisfies O.
Definition The set of predictive explanations of O relative to w is
Expl (O; w) = min
e E : e(w)
P
f
2
w
O
k
kg
An event e is a predictive explanation of O relative to w iff e 2 ExplP (O; w). If ExplP (O; w) =
;
, we say that O is not predictively explainable relative to w.
The distinction between weak and predictive explanations is very similar to that made between
consistency-based diagnosis [23] and predictive (or abductive) diagnosis [22]. This distinction is
e and f are nondeterministic events. Event e predictively explains O,
while f weakly explains O but does not predictively explain O. We are interested here in weak
illustrated in Figure 2. Both
explanations, for these seem most appropriate when dealing with nondeterministic events. However,
we note the following:
Proposition 1 If e is a deterministic event, then e weakly explains O iff e predictively explains O.
Corollary 2 If
EM
is a deterministic event model,
explainable.
11
O is weakly explainable iff O is predictively
To appear, Artificial Intelligence, 1995
O
e
e
O
f
f
w
~O
O
Figure 2: Weak and Predictive Explanations
For a particular world w, Expl(O; w) denotes those most plausible events that could cause O to
be true. The possibilities admitted by such a set of explanations are the possible results of each of
these events. To determine these we simply evolve or progress w in accordance with these possible
event occurrences; that is:
Definition The progression of world w given observation O is the set of worlds
Prog (w O) =
j
[ e(w)
f
O : e Expl(O; w)
\ k
k
2
g
Note that if O is unexplainable relative to w, then Prog (wjO) = ;. This means that there is no event
(among those specified in the model) that could have caused w to evolve into a world that satisfies O.
The occurrence of O relative to w is impossible. We also note that if we are restrict our attention to
predictive explanations, or to deterministic event models, we can rewrite this definition as
Prog(w O) =
j
[ e(w) : e
f
2
Expl (O; w)
g
P
Taking a cue from the Katsuno-Mendelzon update semantics, the progression of a knowledge base
KB given a particular observation O is obtained by considering all plausible evolutions of each world
w
KBk. However, if
2 k
O is unexplainable for some w
2 k
KBk, we take
O to be unexplainable
relative to KB as a whole.
Definition The progression of KB given observation O is the set of worlds
Prog(KB O) =
j
If Prog (wjO) = ; for some w
[ Prog(w O) : w
f
j
KBkg
2 k
KBk, we let Prog (KBjO) = ;.
2 k
The motivation for this last condition, that
O must be explainable relative to every w
12
KBk,
2 k
To appear, Artificial Intelligence, 1995
comes from the KM update semantics itself. In the KM theory of update, the updated KB is constructed
by considering the possible evolution of every possibility admitted by KB. We duplicate this intuition
by considering the progression function of every world in kKBk. If no such evolution is possible for
one of these worlds, we trivialize the result of updating KB. We might have allowed the progression of
KB to be nontrivial even if some worlds could not evolve so as to satisfy O, and define Prog (KBjO)
without this last condition. In other words, we might have considered
Prog(KB O) to be as in the
j
definition, but simply accept, when Prog (wjO) = ;, that w contributes nothing to the construction of
Prog(KB O). However, we adopt the current approach for two reasons. First, our goal is to pursue the
j
analogy with the KM update semantics. Our definition is a direct adaptation. Second, dropping this
restriction has implications for the relationship between belief revision and update. Simply excluding
worlds whose progression is empty is, in effect, performing revision in addition to update. While this
is generally a good idea, the correct way to bring together revision and update requires more drastic
changes in the way update is performed. We elaborate on this in Section 4.
We take progression of KB to be the semantic counterpart of the update of the theory KB. With
such a progression function, we can now define the explanation-change operator relative to a given
event model, which determines the consequences of adopting an observation.
Definition The explanation-change operator induced by an event model EM is EM :
KB EM
O = A LCPL : Prog(KB O) = A
f
2
j
j
g
In our example, we have two event types, Promotion and Raise. A PROMOTION event (promotion
of one level) ensures an employee’s rank is increased and his salary is raised $10,000. Events
RAISE(5) and RAISE(10) raise salary $5000 and $10,000, respectively. We assume the following
event orderings for each department:
Purchasing: RAISE(10)
Finance: RAISE(5)
PROMOTION
PROMOTION
RAISE(5)
RAISE(10)
This is illustrated in Figure 3, where shorter event arcs depict more plausible occurrences. The
explanation relative to purchasing is a raise, while for finance it is a promotion. The updated KB0 is
determined by w0 and v 00 and induces the beliefs described earlier.
As another example, imagine that a warehouse control agent expects a series of trucks to pickup
and deliver certain shipments, but at time t1 an expected truck A has not arrived. Assume that this
might be explained by snow on Route 1 or a breakdown. If snow is the most plausible of the two
13
To appear, Artificial Intelligence, 1995
Prom
R(10)
w’
w
w’’
Prom
R(5)
v
v’’
v’
KB’
KB
Figure 3: An Event Ordering
events, the agent might reach further conclusions by predicting the consequences of that event; for
example, trucks B and D will also be delayed since they use the same route. The proper explanation
and subsequent predictions are crucial, for they will impact on the agent’s decision regarding staffing,
scheduling and so on. Notice also that such explanations are defeasible, which is reflected in the
defeasibility of update: if A is late but B is on time, then snow is no longer plausible (therefore, e.g.,
D will not be delayed).
Finally, we can formalize our initial example. We first adopt a conditional STRIPS-like representation of events, using variables to schematically capture a set of propositions, and take each event
specification to induce the obvious transformation on possible worlds (see, e.g., [18, 6]). We have
two possible events, RAIN and SPRINKLER, with effects as follows:
Event
Condition
RAIN
On(grass,x)
Wet(x), Wet(grass), Wet(patio)
On(patio,x)
Wet(x), Wet(grass), Wet(patio)
else
SPRINKLER
On(grass,x)
else
Effect
Wet(grass), Wet(patio)
Wet(x), Wet(grass)
Wet(grass)
We also have the proposition O asserting that it is overcast, and influencing the plausibility of these
14
To appear, Artificial Intelligence, 1995
two events. A plausibility ordering might be given as follows:
If O then RAIN SPRINKLER
If :O then SPRINKLER
RAIN
Our agent’s knowledge base consists of the beliefs
O; :Wet(book) ; On(patio,book)
f
Inside(book) g
:
Given the fact O (overcast), the most plausible explanation for the observation Wet(grass) is
RAIN. The effect is then Wet(book) if On(patio,x) and :Wet(book) if Inside(book).
Note that had it not been overcast, SPRINKLER would have been the most plausible explanation and
our agent would rest assured that her book is dry.
We should remark at this point that the intent of this model is to provide an abductive semantic
model for update, not a computational model. Just as we do not expect actions or events to be
represented as abstract functions between worlds, explanations will not typically be generated on
a world by world basis. Usually, the same event will explain an observation for a large subset of
the worlds within kKBk. In particular, we expect that kKBk to be partitioned according to some
small number of propositions (or conditions) for which a certain event is deemed to be a reasonable
explanation. Indeed, these can naturally be viewed as conditional explanations, for example, “If
Fred is in Finance, a PROMOTION must have occurred; but otherwise a RAISE must have occurred.”
How such conditional explanations should be generated will be intimately tied to the action or event
representation chosen, and is beyond the scope of this paper.
3.3
Relationship to The Katsuno-Mendelzon Theory
We are interested in the question of whether the explanation-change operator satisfies the KM update
postulates. This is not the case given the formulation above.
Proposition 3 Let
EM
EM
be the explanation-change operator induced by some event model. Then
satisfies postulates (U1), (U4), (U6) and (U7).
There are two reasons why the remainder of the postulates are not satisfied in general, hence two
assumptions that can be made to ensure that EM is an update operator.
The first difference in the explanation-change operator is reflected in the failure of (U2), which
asserts that KB A is equivalent to KB whenever KB entails A. A simple example illustrates why this
15
To appear, Artificial Intelligence, 1995
cannot be the case is general. Consider a KB satisfied by a single world w where w
= A.
j
Postulate
(U2) requires that the observation of A induce no change in KB. However, it may be that the most
plausible event in the ordering w is e, where e(w) =
v = A, then KB
v
f g
for some distinct world v . But assuming
A is captured by v and is thus distinct from w. In order to conform to postulate
(U2), we must make the assumption that no change in w is more plausible than change induced by
j
EM
some event. Formally, we postulate null events and make these most plausible.
Definition The null event is an event n, where n(w) = fwg for all w 2 W .
Definition Let EM
= W; E; be an event model. EM is centered iff the null event n E and,
e.
for each w W and e E (e = n) we have n
h
i
2
2
2
6
w
Thus, a centered event model is one in which the null event is the most plausible event that could
occur at any world. This seems to be the crucial assumption underlying postulate (U2).
Proposition 4 Let EM be the explanation-change operator induced by some centered event model.
Then EM satisfies postulates (U1), (U2), (U4), (U6) and (U7).
This assumption of persistence of the truth of KB seems to be reasonable in many domains, but should
probably be called into question as a general principle. It may be the case in a domain where change
is the norm that, despite the fact that an observation is already believed, some change in KB should
be forthcoming. As an example, consider an agent monitoring a control system producing some
product. It observes a display that indicates whether the system is proceeding normally. If it believes
that a normal condition is displayed before the next observation, observing that the display (still)
indicates normal should not require that its other beliefs not change: it may, for instance, update the
number of units produced in response to this observation. In this sense, the more general nature of
the explanation-change operator may be desirable.
Postulate (U3) is also violated by our model, and for a similar reason, so too are (U5) and (U8).
For a given KB, we may have that Prog (wjO) = ; for each w
KBk. In other words, there are no
2 k
possible events that would cause an observation O to become true. The potential for such unexplainable
observations clearly contradicts (U3), which asserts that KB O must be consistent for any consistent
O. The assumption underlying (U3) in update semantics seems to be the following: every consistent
proposition is explainable, no matter how unlikely. In order to capture this assumption, we propose a
class of event models called complete.
EM = W; E; be an event model. EM is complete iff for each
proposition O and w W , O is explainable relative to w.
Definition Let
h
i
2
16
consistent
To appear, Artificial Intelligence, 1995
Proposition 5 If
EM is a complete event model then Prog(KB O) =
j
6
;
for any consistent
O and
KB.
This condition is sufficient to ensure (U5) and (U8) are satisfied as well.
Proposition 6 Let EM be the explanation-change operator induced by some complete event model.
Then EM satisfies postulates (U1), (U3), (U4), (U5), (U6), (U7) and (U8).
The completeness of an event model refers, in fact, to the completeness of its event set
E.
If
this set is rich enough to ensure that, for every world and observation, some event can make that
observation hold, then the event model will be complete. Typically, domains will not be so wellbehaved. However, the simple addition of a miracle event to an event set will ensure completeness.
Intuitively, a miracle is some event which is less plausible than all others and whose consequences
are entirely unknown.
Definition Let EM
= W; E; be an event model. A miracle is an event m such that m(w) = W
for all w W , and e
m for all w W and e E (e = m).
h
2
Proposition 7 Let EM
i
w
= W; E;
h
2
i
2
6
be an event model. If E contains a miracle event, then EM is
complete.
If all observations must be explainable, and no observation is permitted to force an agent into
inconsistency, then miracles are one embodiment of the required assumptions. The reasonableness
of such a requirement can be called into question, however. Having unexplainable observations is,
in general, a natural state of affairs. Rather than relying on miraculous explanations, the threat of an
inconsistency can force an agent to reconsider the observation, its theory of the world, or both. As
we will see in the concluding section, it is just this type of inconsistency that can force an agent to
revise its beliefs about the world prior to the observation. Update postulate (U3) makes it difficult to
combine update with revision in this way.
If we put together Propositions 4 and 6, we obtain the main representation result for explanationchange.
Theorem 8 Let EM be the explanation-change operator induced by some complete, centered event
model. Then EM satisfies update postulates (U1) through (U8).
A useful perspective on the relationship between explanation change and update comes to light
when one considers that the plausibility ordering on events quite naturally induces an indexed family
of preorders of the type required in the Katsuno-Mendelzon update semantics.
17
To appear, Artificial Intelligence, 1995
Definition Let EM
each w
= W; E;
h
i
be an event model. The plausibility ordering induced by EM , for
W , is defined as follows: v u iff for any event e
e.
some event e (where v e (w)) such that e
2
w
2
v
u
v
v
w
such that u
2
e (w), there is
u
u
Intuitively, the more plausible some causing event ev for world v is (relative to w), the more plausible
evolution v of world w is deemed to be (according to w ).
w W
centered event model EM . Then
Theorem 9 Let
fw
:
2
g
be the family plausibility orderings induced by some complete,
(a) Each relation w is a faithful preorder over W .
(b) The change operation determined by fw : w 2 W g is a KM update operator.
(c) The update operator determined by fw :
w W
2
g
is equivalent to the explanation-
change operator EM .
We note that if the event model is not centered then the generated preorder is not necessarily faithful.
If the model is not complete, then we have only a restricted form of faithfulness. It will be the case
w < v for any world v that is a possible evolution of w (i.e., if v e(w) for some e E ).
However, those worlds that cannot result from the application of some event to w will be unrelated
is faithful relative to the connected component of
that
to w. In this case, we can say that
includes w. Intuitively, we want to ignore those worlds that are not “reachable” from w. To do this
that
2
w
w
2
w
we can simply define an update operator using the relation w restricted to such worlds:
v : v e(w) for some e E
f
2
2
g
This ensures that unrelated worlds are not trivially minimal in the ordering relation w .
If we have an event model where each event ordering is a total preorder, then the induced
plausibility orderings over worlds are also total preorders.
Proposition 10 Let
EM = W; E;
h
i
be a complete event model such that w is a total preorder
for each w 2 W . Then each plausibility ordering w induced by EM is a total preorder.
The circumstance where a set of events is totally preordered by plausibility may arise rather frequently;
for instance, events may be ranked according to some integer scale, or assigned some qualitative
probability ranking. Therefore, the properties of such total update operators are of interest. We can
18
To appear, Artificial Intelligence, 1995
extend the Katsuno-Mendelzon representation theorem to deal with update operators of this type.
The required postulate embodies a variant of the principle of rational monotonicity, cited widely in
connection with nonmonotonic systems of inference and conditional logics (see, e.g., [3, 19]).
(U9) If KB is complete, (KB A) 6j=
B and (KB A) = C then (KB (A B)) = C
:
j
^
j
Theorem 11 An update operator satisfies postulates (U1) through (U9) iff there exists an appro-
priate family of faithful total preorders fw : w
2
W
g
that induces (in the usual way).
Corollary 12 Let EM be the explanation-change operator induced by some total order event model.
Then EM satisfies postulate (U9).
Indeed, Katsuno and Mendelzon [16] also discuss the possibility of totally ordered plausibility rankings
and provide a postulate (U9) related to the one above, and the proof of equivalence is similar to that
suggested by them (see also their work on total orders for belief revision [17]).
As a final remark, we note that the converses of Theorems 8 and 9 are trivially and uninterestingly
true. For any update operator , one can construct an appropriate set of events (and orderings) that
will induce that operator. This not of interest, since the point of explanation-change is to provide a
natural view of update, characterizable in terms of the events of an existing domain. The ability to
construct such events to capture a particular update operator provides little insight into update. The
appropriate perspective is to reject any update operator (in a given domain) that cannot be induced by
the existing set of events (or event model).
4
Concluding Remarks
We have provided an abductive model for incorporating into an existing belief set observations that
arise through the evolution of the world. While our model allows more general forms of change than
KM update, we can impose restrictions on our model to recover precisely the KM theory. However,
these restrictions are inappropriate in many cases, calling into question the suitability of some of the
update postulates.
4.1
Relationship to Belief Revision
It has frequently been suggested that abduction can be modeled by appeal to belief revision [12].
Boutilier and Becher [5] present a model of abduction along these lines, whereby an explanation for
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To appear, Artificial Intelligence, 1995
O, with respect to some KB, is a sentence E such that KB E entails O.
words, had an agent believed the explanation E it would have believed the observation O.
an observation
In other
The abductive view of update suggests that update may also be viewed as a form of belief revision.
Since explanations take the form of event occurrences, an interpretation using belief revision takes
update to be the process of an agent revising its beliefs about whether an event occurred and just what
that event was. For instance, in our main example the observation Sal(50000) can be viewed as
causing an agent to give up its belief that no event has occurred (i.e., the world has not changed) and
accept the fact that something has happened — in particular, it accepts the belief
Dept(P)
Occurred(RAISE(10) ) ^ Dept(F)
Occurred(PROMOTION )
Of course, we have not provided an explicit logical language for the representation of actions or
events, and in particular, have not provided a method for revising beliefs about such occurrences.
However, there are a number of ways to model this type of belief revision, including using histories
or runs of a system as our basic semantic objects. A run is essentially a sequence of world states
capturing a particular evolution of a system. Using these as semantic primitives one can capture
beliefs about the actual state of the world in addition to event occurrences. While not directly suited
to our task, the revision model of Friedman and Halpern [11], in which runs are ranked in manner
suitable for belief revision, is precisely the type of system upon which a more elaborate model of
update, revision and explanation can be built.
When viewed in this way, certain problems with the update model, as formulated by Katsuno and
Mendelzon and recast here, become apparent. The types of explanations one is willing to consider
are restricted to event occurrences. In other words, an agent is bound to revise its beliefs only about
possible event occurrences and their consequences. Thus, an agent making an observation is not
allowed to entertain the possibility that its knowledge base KB was incomplete or incorrect. It can
only change its beliefs about the post-event world state. Semantically, this restriction is apparent in
our definition of update (as well as Katsuno and Mendelzon’s). We require that every w
KBk be
2 k
progressed according to likely explaining events.
It is just this restriction that calls into question the suitability of update as a “stand-alone” belief
change operator. Of particular concern, as emphasized earlier, is postulate (U3). This embodies the
assumption that all observations are explainable in terms of some event. This is not always reasonable.
For instance, in our database example we might have a transaction to update Fred’s salary to $90,000
when there is a salary cap of $80,000 in Finance. Thus, no event could have caused this salary change
20
To appear, Artificial Intelligence, 1995
if Fred is indeed in Finance. Far from being a miraculous occurrence, it suggests that Fred in actually
in Purchasing. Thus the observation not only forces KB to be updated (reflecting a change in the
world), but also revised (reflecting additional knowledge about the world).
Note that this is not an artifact of out definition of update, One might argue that we should
simply update those worlds for which explanations exist and ignore the others. This minor adjustment
seems reasonable, but it is no longer simply update. Rather it is a combination of update and revision.
Furthermore observations may often be unexplainable for every world in kKBk. For instance, suppose
a solution is believed to be an acid, when a litmus strip is dipped into it and promptly turns blue. This
is not explainable for any KB-world (it should turn red) in terms of event effects. As such, the minor
adjustment of our definition of update is not sufficient. We may want to update KB consistently in
circumstances where no possible event could give rise to the observation given our current state of
belief. Instead, the intuitive explanation in this example consists of two parts: the first postulates that
the event of dipping the paper in the solution occurred; the second suggests that the solution is in fact
a base. This requires revision of KB — we must change our beliefs about the pre-event state of the
world in order to modify KB correctly.
Finally notice that an observation need not be strictly unexplainable to force revision. Often
an implausible explanation will suffice. For instance, a raise to $90,000 might not be impossible in
Finance, but just so implausible that the database is willing to accept the fact that Fred is in Purchasing.
To adequately reflect such considerations, we must have the ability to compare the plausibility of event
occurrences with the plausibility of beliefs about the world state. This provides further support for
more expressive models and languages in which event occurrences can be reasoned with explicitly.
Issues of this sort make postulate (U3) (and certain aspects of (U5) and (U8)) somewhat questionable, and provides motivation for adopting an abductive view of update. This perspective is
especially fruitful when combining the process of update (changing knowledge) with belief revision
(gaining knowledge). A model that puts both components together in a broader abductive framework
is described in [2]. Roughly, the logic for belief revision set forth in [4] is used to capture the revision
process, but is combined with elements of dynamic logic [14] to capture the evolution of the world
due to action occurrences.
4.2
Related Work
Other have presented models of update that, like ours and unlike the KM-model, have their basis
in reasoning about action. del Val and Shoham [8, 9], using the situation calculus, show how one
21
To appear, Artificial Intelligence, 1995
can determine an update operator by reasoning about the changes induced by a given action. Very
roughly, when some KB is to be updated by an observation O, they postulate the existence of some
B
whose predicted effects, when applied to the “situation” embodied by KB, determine the
action AK
O
form of the update operator. Most critically, the effect axiom for such an action states that O holds
B
is applied to KB, and other effects are inferred via persistence mechanisms.
when AK
O
This model differs from ours in a number of rather important ways. First, del Val and Shoham
assume that the update formula O describes the occurrence of some action or event. This severely
restricts the scope of update, which in general can accept arbitrary propositions. They provide no
mechanism for explaining an observation using the specification of existing actions. In order to deal
with arbitrary observations an action is “invented” for the purpose of causing any observation in any
situation. Naturally, the effects of such new actions are not specified a priori in the domain theory.
So they propose that the effect of invented actions is to induce minimal change in the knowledge
base according to some persistence mechanism. However, the plausible cause of an observation O
may carry with it, in general, other drastic (rather than minimal) changes in KB. This can only be
accounted for by explaining an observation in terms of existing actions. A persistence mechanism is
required primarily because existing action or event specifications are not employed.
Another drawback of this model is its failure to account for the possibility that any of a number
of actions might have caused
O, and that update should reflect the most plausible of these causes.
Finally, there is an assumption that the update of KB is due to the occurrence of a (known) single
action. As we have described above, this will usually not be the case. Conditional explanations,
explanations that use different actions for different “segments” of KB, will be very common.
A related mechanism is proposed by Goldszmidt and Pearl [13], who use qualitative causal networks to represent an action theory. Again, update formula are implicitly assumed to be propositions
asserting the occurrence of some action or event. An observation
O is incorporated by assuming
some proposition do(O) has become true, and using a forced-action semantics to propagate its effects.
Explanations are not given in terms of existing actions.
We should point out that both proposals adopt a theory of action that provides a representation
mechanism for actions and effects, as well as incorporating a solution to the frame problem (implicitly
in the case of Goldszmidt and Pearl). We have side-stepped such issues by focusing on the semantics
of update. We are currently investigating various action representations, such as STRIPS and the
situation calculus, and the means they provide for generating conditional explanations. This is partially
developed in [2], where we provide a representation for actions using a conditional default logic to
capture the defeasibility and nondeterminism of action effects, and use elements of dynamic logic to
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To appear, Artificial Intelligence, 1995
capture the evolution of the world. Action theories such as those exploited in [8, 13] might also be
used to greater advantage.
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August 1990.
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database and revising it. In Proceedings of the Second International Conference on Principles
of Knowledge Representation and Reasoning, pages 387–394, Cambridge, 1991.
[17] Hirofumi Katsuno and Alberto O. Mendelzon. Propositional knowledge base revision and
minimal change. Artificial Intelligence, 52:263–294, 1991.
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Acknowledgements
Discussions with Ray Reiter have helped to clarify my initial thoughts on update. Thanks to Richard
Dearden and David Poole for helpful discussions on this topic and to Alvaro del Val and David
Makinson for well-considered comments on an earlier draft of this paper. Thanks also to the referees
for suggestions that helped clarify the presentation. This research was supported by NSERC Research
Grant OGP0121843.
A
Proofs of Main Results
Proposition 3 Let
EM
EM
be the explanation-change operator induced by some event model. Then
satisfies postulates (U1), (U4), (U6) and (U7).
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To appear, Artificial Intelligence, 1995
Proof Assume an event model M
= W; E;
h
i
and associated update operator (for simplicity we
drop the subscript). We show in turn that each of these postulates is satisfied.
(U1) By definition
have KB A j= A.
Res(A; w)
A
k
k
for all
w (hence for all w
KBk). Immediately we
2 k
A B. Then e explains A w.r.t KB iff e explains B , for any event e. Thus,
KB A KB B .
B and
(U6) Suppose KB A = B and KB B = A. Then we have Res(A; KB)
Res(B; KB) A . If v Res(A; KB), then for some w KB we have a most plausible
explaining event e for A such that v e(w). However, since v
B , e must also be a most
plausible explaining event for B as well; otherwise there must exist some more plausible event
f e that explains B, contradicting the fact that e is most plausible for A (since Res(B; KB)
A ). Therefore, v Res(B; KB), so Res(A; KB) Res(B; KB). By symmetry the reverse
containment holds, so Res(A; KB) = Res(B; KB) and we have KB A KB B .
(U7) Let KB be complete so that KB = w for some world w. Suppose v Res(A; w)
Res(B; w). (If there is no such v then (KB A) (KB B) is inconsistent and (U7) holds
trivially.) Then there is some e such that v e(w) and e is a most plausible explaining event
for both A and B . This ensures that e explains A B and that v Res(A B; w). Hence,
Res(A; w) Res(B; w) Res(A B; w). Therefore, (KB A) (KB B) = KB (A B).
(U4) Suppose j=
k
k
j
j
k
2
2 k
2
k
k
2 k
k
k
k
2
k
k
f
g
2
^
\
2
_
\
2
_
^
_
j
_
Proposition 4 Let EM be the explanation-change operator induced by some centered event model.
Then EM satisfies postulates (U1), (U2), (U4), (U6) and (U7).
Proof Given Proposition 3, we need only show that (U2) is satisfied by centered event models.
Assume that
M = W; E;
h
i
is such a model, inducing update operator (for simplicity we
drop the subscript). Suppose KB j= A. Then for each w
KBk, we have w j= A; and the null
2 k
event is the most plausible explaining event for each such world. Thus Res(A; KB)
and KB A is equivalent to KB.
=
k
KBk
Proposition 6 Let EM be the explanation-change operator induced by some complete event model.
Then EM satisfies postulates (U1), (U3), (U4), (U5), (U6), (U7) and (U8).
Proof Given Proposition 3, we need only show that (U3), (U5) and (U8) are satisfied by complete
models. Assume that
M = W; E;
h
i
is such a model, inducing update operator
simplicity we drop the subscript).
26
(for
To appear, Artificial Intelligence, 1995
(U3) Since M is complete, any satisfiable A is explainable for every w. So if KB is satisfiable,
Res(A; KB) = and KB A is satisfiable.
(U5) Let v Res(A; KB)
B (if there is no such v , then (U5) holds trivially). Then for
KB there is a most plausible explanation e for A (w.r.t. w) such that v e(w).
some w
This event e must also be a most plausible explanation for A B w.r.t. w (otherwise some more
plausible event would explain A B , hence A). Thus v Res(A B; KB). Notice that since
A B must be explainable for every world, we cannot have that Res(A B; KB) is set to the
empty set. Thus, Res(A; KB)
B Res(A B; KB) and (KB A) B = KB (A B).
(U8) Assume that KB1 KB2 is satisfiable. We have w Res(A; KB1 KB2) iff there is some
v KB1 KB2 such that w Res(A; v ). Such a v is either in KB1 or KB2 , so this holds
iff w Res(A; KB1) Res(A; KB2 ). Therefore, (KB1 KB2 ) A (KB1 A) (KB2 A).
6
;
2
\ k
2 k
k
k
2
^
^
2
^
^
^
\k
k ^
_
2 k
_
2
k
k
[
_
w W
centered event model EM . Then
Theorem 9 Let
fw
:
2
g
j
^
_
2
2
^
k
k
k
_
be the family plausibility orderings induced by some complete,
(a) Each relation w is a faithful preorder over W .
(b) The change operation determined by fw : w 2 W g is an update operator.
(c) The update operator determined by fw :
w W
2
g
is equivalent to the explanation-
change operator EM .
Proof Let w be some world in a complete, centered event model EM
= W; E;
h
, and let w be
i
an induced ordering.
(a) That
w
is reflexive and transitive follows immediately from the definition w in
terms of the event ordering w and the fact that w is itself reflexive and transitive.
Hence w is a preorder. Since null action n w
e for all e = n and n(w) = w , we
have w
v for all v = w (if v e (w) for some event e ). Thus is faithful.
Finally, since EM is complete, for any v there is some e such that v e (w). Thus
w
6
2
6
v
v
v
w
f
g
w
2
v
is persistent.
(b) The representation theorem of Kastuno and Mendelzon ensures that the family of
orderings fw : w 2 W g generates an update operator satisfying (U1) through (U8).
27
To appear, Artificial Intelligence, 1995
(c) Denote by
the update operator generated by
fw
:
w W
2
. We will show that
g
KB A KB EM A for any consistent KB and A. Assume v
v 2 kKB Ak iff v 2 [w2kKBk fminw fv : v j= Agg
iff
iff
for some w
for some w
iff
iff
. We have
k
KBk, if u w v , then u 6j= A
2 k
KBk and event e, v
2
e(w) and
for all e0
e, we have e0(w) A =
for some w
KB , v Res(A; w)
v Res(A; KB)
v KB A
\k
k
2 k
w
iff
A
2 k
2 k
k
;
2
2
2 k
k
Proposition 10 Let
EM = W; E;
h
i
be a complete event model such that w is a total preorder
for each w 2 W . Then each plausibility ordering w induced by EM is a total preorder.
Proof Theorem 9 ensures that w is a preorder. For any world u, let
e
u
denote any event that has
u relative to w; i.e., u e (w) (such an event must exist since EM is complete).
u. Then there must be some e such that for all
Consider two worlds u and v . Suppose v
e ,e
e . Since is a total preorder, e
e for all such e ; and u v. Thus, is
a total preorder. outcome
2
u
6w
v
v
6w
u
w
u
u
6w
v
v
w
w
Theorem 11 An update operator satisfies postulates (U1) through (U9) iff there exists an appro-
priate family of faithful total preorders fw : w
2
W
g
that induces (in the usual way).
Proof We first assume a suitable family of preorders. The representation result of Katsuno and
Mendelzon ensures that the induced update operator
satisfies (U1) through (U8). We now
show that it also satisfies (U9). LetKB be complete with kKBk = fwg. Suppose KB A 6j= :B
and KB A j= C . Let min(A) denote the set minw fv : v
and min(A) \ kB k =
6
;. Since w is a total preorder,
=A
j
. Then we have min(A) kC k
g
min(A ^ B ) min(A) \ kB k kC k
so KB (A ^ B ) j= C . Therefore (U9) is satisfied.
Now we suppose satisfies postulates (U1) through (U9). To prove that a suitable family of
orderings exist, we adopt the basic technique of Katsuno and Mendelzon [15]. However, the
28
To appear, Artificial Intelligence, 1995
orderings are constructed in a rather different fashion to ensure that the preorders are total. As
preliminary notation, for any set of worlds X , we write X to denote some sentence such that
= X . To emphasize that this will be the object of revision, we write KBf g instead of
f g . We note that (U1) and (U3) ensure that KBf g 0 for some X 0 X . We will
X . We can now define a family of
also make use of the fact that 0 = for any X 0
k
Xk
w
w
w
j
X
X
X
X
ordering relations based on as follows:
v
u
w
iff
v
2 k
KBfwg fv;ug k
We first show that w is a faithful persistent preorder. Clearly, w is reflexive, since (U1) and
(U3) ensure that v
so either v
w
KBfwg fvgk. Similarly, at least one of v or u is in kKBfwg fv;ug k,
2 k
u or u
v . It is easy to verify that
w
w
is faithful and persistent due to (U2)
and (U3). It simply remains to verify that w is transitive. So suppose v
This ensures that v
KBfwg fv;ug k and u 2 kKBfwg fu;tgk.
f g .
Then KBf g f
g = f g and KBf g f g = f g.
By (U9), KBf g (f
g f g) = f g,
or equivalently KBf g f g = f g.
KBf g f g .
But this contradicts the fact that u
Suppose KBf g f
g = f g and KBf g f g = f g .
Then KBf g f
g = f g and KBf g f g = f g.
By (U9), KBf g (f
g f g ) = f g ,
or equivalently KBf g f g = f g.
KBf g f g and v
f g
But this contradicts the fact that v
Thus, if u
KBf g f
g then v KBf g f g .
w
w
6j
v;u;t
w
t
:
u;t
^
v;u;t
w
w
u;t
j
u;t
j
w
w
6j
v;u;t
6j
v;u;t
v;u;t
w
j
w
v;u;t
j
t
k
k
v;u;t
v;u;t
j
:
j
v
u;t
u;t
u;t
2 k
2 k
w
v;u
j
u;t
w
v;u
v;u
w
u
^
v;u;t
t
:
:
t
2 k
w
u and u
2 k
(a) Suppose KBfwg fv;u;tg
(b)
w
w
2 k
v;u
w
k
62 k
v;u;t
u;t
k
.
k
By (a) and (b), we know that KBfwg fv;u;tg is not equivalent to ftg, fug or fu;tg.
Thus, v
KBfwg fv;u;tgk.
2 k
Now by (U5) we have (KBfwg fv;u;tg) ^ fv;tg
Since v
= KBf g f g.
g , this conjunction is consistent, thus v
KBfwg fv;u;t
2 k
Therefore v
w
t and
w
j
w
v;t
2 k
k
is transitive.
29
KBfwg fv;tgk.
t.
To appear, Artificial Intelligence, 1995
Finally, we demonstrate that
[
KB Ak =
k
w
f
2kKBk
minfv : v
w
=A
j
gg
The remainder of the proof follows closely that of Katsuno and Mendelzon, but we include it for
completeness. We assume KB and A are consistent, for the relations holds trivially otherwise.
We first show that this relation holds for any complete KB. Assume kKBk
min(A) to denote the set of minimal A-worlds in w .
Suppose v
= w
f
. We use
g
= KB A but v min(A). Then there is some u < v such that u = A. By (U5),
(KB A) f g = KB f g ; and by definition of , KB f g f g . But then
v = KB A. So v must be in min(A). Thus, KB A min(A).
Now suppose v min(A). Let A = u1; ; u . We have that
6j
j
^
v;u
62
j
w
v;u
k
2
k
A
And since v
u for each i
i n. That is, v satisfies
i
j
w
k
f
f g
v;u1
n (since v
2
v;u
u
k ng
__
f
v;un
g
min(A)), we have v
2 k
KB fv;ui g k for each
(KB f g )
v;u1
^^
(KB f
v;un
g)
By (U7), v therefore satisfies
KB (fv;u1 g _ _ fv;un g)
That is, v
2 k
KB Ak. Therefore, min(A) kKB Ak.
The result holds for any complete KB. However, any KB is equivalent to the disjunction of
some finite set of complete KBs. Thus, by (U8) we have
[
KB Ak =
k
w
f
2kKBk
30
minfv : v
w
=A
j
gg